Question

The range of $f\left(x\right)={\cos }^{-1}x+2{\tan }^{-1}x+3{\mathrm{cosec}}^{-1}x$ is

• A. $[-\pi ,2\pi ]$
• B. $\varphi$
• C. $(-\pi ,2\pi )$
• D. $\{-\pi ,2\pi \}$

Answer 1

The correct option is D $\{-\pi ,2\pi \}$

Domain of ${\cos }^{-1}x$ is $[-1,1]$
Domian of ${\tan }^{-1}x$ is $R$
Domain of ${\text{cosec}}^{-1}x$ is $(-\infty ,-1]\cup [1,\infty )$
$\therefore$ Domain of $f\left(x\right)$ is $[-1,1]\cap R\cap \left(\right(-\infty ,-1]\cup [1,\infty \left)\right)=\{-1,1\}$

$f(-1)=\pi +2(-\frac{\pi }{4})+3(-\frac{\pi }{2})=-\pi$
$f\left(1\right)=0+2\left(\frac{\pi }{4}\right)+3\left(\frac{\pi }{2}\right)=2\pi$
$\therefore$ Range $=\{-\pi ,2\pi \}$